Friday, July 8, 2011

Theory Update 91

Now that some people have realised that M Theory is all about triple braids and ribbons, let us recall yet again the accurate MINOS results, indicating a difference between $\nu$ and $\overline{\nu}$ masses, as shown by the Bilson-Thompson $B_3$ ribbons when augmented with mirror diagrams. The Koide mass matrix parameter $6/27 \pm \pi/12$ contains a $2/9$, reflecting the centre qutrit paths on a tetractys M5 brane.

1. Ironically, I have discovered a small convergence between mainstream string phenomenology and what you are proposing. Lubos recently highlighted a construction which gets the MSSM from branes placed on a singular dP3 surface in a Type IIB string space-time. Del Pezzo surfaces are the subject of the "mysterious duality" paper, in which you found the line diagram resembling the tetractys. One quasi-conventional hope regarding this duality, by the way, is that the d=4 surfaces themselves might supply a purely 4-dimensional description of M-theory; see here, and also here for historical background (section 1.1) and a possible application to the twistor string. Lubos also remarks that this dP3 phenomenology paper is in the part of the string landscape which looks like the real world, and which can be approached in various ways by F-theory, heterotic strings, etc. From this I conclude that, if the "mysterious" reformulation of M-theory exists, all those other semirealistic vacua may also have a formulation similar to, e.g., "N=(2,1) strings on dP3".

Of course, all this mainstream string phenomenology supposes supersymmetry, and implements that in terms of unobserved heavy superpartners. But as a fan of the Rivero correspondence, I would like to think that supersymmetry is already enfolded within the standard model. One might hope that even this nonstandard realization of supersymmetry, embedded in string theory, would lie in this same vicinity of the dP3 model.

Now we come to the point. The toric diagram for a dP3 is just a triangle with the corners cut off (points "blown up" and replaced by curves), not too far from the plain triangle which corresponds to dP0 and which, with a cubic curve added, supplied the diagram resembling your tetractys dual. And here you are getting your Koide mass matrices from such structures.

2. Yes, kneemo has already figured this out, and much more.

3. Recall also that toric geometry is the subject of the green book, from which the associahedra arise as secondary polytopes. This will give matrices a nice entropic interpretation. We have been thinking about all this for some time now. My supersymmetry, as you know, involves the quantum Fourier transform, on braid diagrams, and everything comes from braids, absolutely everything. This is the point of my new paper.

4. And apologies for mixing up M5 and M8 branes before. From this point of view the dimension of the brane is just, well, whatever it is. It is the qutrit path diagram that is fundamental.

5. Actually, there is a misconception in what I wrote. dP3 is "mysteriously dual" to M-theory compactified on T^4. That is the background which one might imagine was equivalent to (maybe) N=(2,1) strings on dP3. But the phenomenology paper linked by Lubos is about something quite different; you could say it's about Type IIB strings on "dP3-branes", D-branes in a conical space with a dP3 "tip". They are quite different geometries.

However, the comparison did at least give me some further insight into what you might be doing. First (using some NCG jargon) I thought in terms of an "almost commutative twistor space". That is, a space in which the macroscopic dimensions exhibit twistorial nonlocality (as in BCFW), multiplied at each point by a noncommutative or nonassociative spectral geometry, as in the noncommutative standard model.

But I assume that ultimately (penultimately?), we're talking about mapping some exotic algebra onto abstract braids, with one part of the algebra giving rise to particles in space-time (thus, Bilson-Thompson), and another part giving rise to quantum numbers and interactions (thus, Koide).

By the way, if kneemo ever specifically takes an interest in Rivero's correspondence, he might want to examine Sultan Catto's implementation of hadronic supersymmetry, which employs split octonions.

6. Oh, don't worry, kneemo knows all about the split octonions. And yes, the almost-NC idea is useful.

And the point is that Spaces and Algebras BOTH arise as abstract braids. One does not start with ad hoc algebras. Everything is canonical. If you read my draft again with this in mind, it will hopefully make more sense. You see, because quantum gravity must create emergent geometry, it must also create emergent algebras. Jordan algebras are close to braids precisely because they are measurement algebras.